EUROPA-TECHNICAL BOOK SERIES EUROPA-TECHNICAL for the Metalworking Trades Trades
Ulrich Fischer
Max Heinzler
Friedrich Näher
Heinz Paetzold
Roland Gomeringer
Roland Kilgus
Stefan Oesterle
Andreas Stephan
Mechanical and Metal Trades Handbook 2nd English edition
Europa-No.:1910X
VERLAG EUROPA LEHRMITTEL · Nourney, Vollmer GmbH & Co. KG Düsselberger Straße 23 · 42781 Haan-Gruiten · Germany
Original title: Tabellenbuch Metall, 44th edition, 2008 Authors: Ulrich Fischer Roland Gomeringer Max Heinzler Roland Kilgus Friedrich Näher Stefan Oesterle Heinz Paetzold Andreas Stephan
Dipl.-Ing. (FH) Dipl.-Gwl. Dipl.-Ing. (FH) Dipl.-Gwl. Dipl.-Ing. (FH) Dipl.-Ing. Dipl.-Ing. (FH) Dipl.-Ing. (FH)
Reutlingen Meßstetten Wangen im Allgäu Neckartenzlingen Balingen Amtzell Mühlacker Kressbronn
Editor: Ulrich Fischer, Reutlingen Graphic design: Design office of Verlag Europa-Lehrmittel, Leinfelden-Echterdingen, Germany The publisher and its affiliates have taken care to collect the information given in this book to the best of their ability. However, no responsibility is accepted by the publisher or any of its affiliates regarding its content or any statement herein or omission there from which may result in any loss or damage to any party using the data shown above. Warranty claims against the authors or the publisher are excluded. Most recent editions of standards and other regulations govern their use. They can be ordered from Beuth Verlag GmbH, Burggrafenstr. 6, 10787 Berlin, Germany. The content of the chapter "Program structure of CNC machines according to PAL" (page 386 to 400) complies with the publications of the PAL Prüfungs- und Lehrmittelentwicklungsstelle (Institute for the development of training and testing material) of the IHK Region Stuttgart (Chamber of Commerce and Industry of the Stuttgart region).
English edition: Mechanical and Metal Trades Handbook 2nd edition, 2010 6 5 4 3 2 1 All printings of this edition may be used concurrently in the classroom since they are unchanged, except for some corrections to typographical errors and slight changes in standards.
ISBN 13 978-3-8085-1913-4 Cover design includes a photograph from TESA/Brown & Sharpe, Renens, Switzerland All rights reserved. This publication is protected under copyright law. Any use other than those permitted by law must be approved in writing by the publisher. © 2010 by Verlag Europa-Lehrmittel, Nourney, Vollmer GmbH & Co. KG, 42781 Haan-Gruiten, Germany http://www.europa-lehrmittel.de Translation: Techni-Translate, 72667 Schlaitdorf, Germany; www.techni-translate.com www.techni-translate.com Eva Schwarz, 76879 Ottersheim, Germany; www.technische-uebersetzungen-eva-schwarz.de Typesetting: YellowHand GbR, 73257 Köngen, Germany; www.yellowhand.de Printed by: Media Print Informationstechnologie, D-33100, Paderborn, Germany
3
Preface
1 Mathematics
M
The Mechanical and Metal Trades Handbook is well-suited for shop reference, tooling, machine building, maintenance and as a general book of knowledge. It is also useful for educational purposes, especially in practical work or curricula and continuing education programs. Target Groups • • • • • • • • •
9 – 32
2 Physics
P
Industrial and trade mechanics Tool & Die makers Machinists Millwrights Draftspersons Technical Instructors Apprentices in above trade areas Practitioners in trades and industry Mechanical Engineering students
33 – 56
3 Technical
TD
drawing
57 – 114
Notes for the user The contents of this book include tables and formulae in eight chapters, including Tables of Contents, Subject Index and Standards Index. The tables contain the most important guidelines, designs, types, dimensions and standard values for their subject areas. Units are not specified in the legends for the formulae if several units are possible. However, the calculation examples for each formula use those units normally applied in practice. Designation examples, which are included for all standard parts, materials and drawing designations, are highlighted by a red arrow ( fi). The Table of Contents in the front of the book is expanded further at the beginning of each chapter in form of a partial Table of Contents. The Subject Index at the end of the book (pages 417–428) is extensive. The Standards Index (pages 407–416) lists all the current standards and regulations cited in the book. In many cases previous standards are also listed to ease the transition from older, more familiar standards to new ones. We have thoroughly revised the 2nd edition of the "Mechanical and Metal Trades Handbook" in line with the 44th edition of the German version "Tabellenbuch Metall". The section dealing with PAL programming of CNC machine tools was updated (to the state of 2008) and considerably enhanced.
4 Material science
MS 115 – 200
5 Machine elements
ME 201 – 272
6 Production Engineering PE 273 – 344
7 Automation and Information Tech345 –406 nology
A
8 International material comparison chart, Standards 407–416
S
Special thanks to the Magna Technical Training Centre for their input into the English translation of this book. Their assistance has been extremely valuable. The authors and the publisher will be grateful for any suggestions and constructive comments. Spring 2010
Authors and publisher
4
Table of Contents 1 Mathematics 1.1
1.2
1.3
1.4
Numerical tables Square root, Area of a circle . . . . . . . . . 10 Sine, Cosine . . . . . . . . . . . . . . . . . . . . . . 11 Tangent, Cotangent . . . . . . . . . . . . . . . . 12 Trigonometric Functions Definitions . . . . . . . . . . . . . . . . . . . . . . . . 13 Sine, Cosine, Tangent, Cotangent . . . . 13 Laws of sines and cosines . . . . . . . . . . . 14 Angles, Theorem of intersecting lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Fundamentals of Mathematics Using brackets, powers, roots . . . . . . . 15 Equations . . . . . . . . . . . . . . . . . . . . . . . . . 16 Powers of ten, Interest calculation . . . . 17 Percentage and proportion calculations . . . . . . . . . . . . . . . . . . . . . . . 18 Symbols, Units Formula symbols, Mathematical symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 19 SI quantities and units of measurement . . . . . . . . . . . . . . . . . . . . . 20 Non-SI units . . . . . . . . . . . . . . . . . . . . . . 22
9 1.5
1.6
1.7
1.8
1.9
2 Physics 2.1
2.2
2.3
2.4
2.5
2.6
Motion Uniform and accelerated motion . . . . . 34 Speeds of machines . . . . . . . . . . . . . . . . 35 Forces Adding and resolving force vectors . . . 36 Weight, Spring force . . . . . . . . . . . . . . . 36 Lever principle, Bearing forces . . . . . . . 37 Torques, Centrifugal force . . . . . . . . . . . 37 Work, Power, Efficiency Mechanical work . . . . . . . . . . . . . . . . . . 38 Simple machines . . . . . . . . . . . . . . . . . . 39 Power and Efficiency . . . . . . . . . . . . . . . 40 Friction Friction force . . . . . . . . . . . . . . . . . . . . . . 41 Coefficients of friction . . . . . . . . . . . . . . 41 Friction in bearings . . . . . . . . . . . . . . . . 41 Pressure in liquids and gases Pressure, definition and types . . . . . . . 42 Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . 42 Pressure changes in gases . . . . . . . . . . 42 Strength of materials Load cases, Load types . . . . . . . . . . . . . 43 Safety factors, Mechanical strength properties . . . . . . . . . . . . . . . . . 44 Tension, Compression, Surface pressure . . . . . . . . . . . . . . . . . . 45 Shear, Buckling . . . . . . . . . . . . . . . . . . . . 46
Lengths Calculations in a right triangle . . . . . . . 23 Sub-dividing lengths, Arc length . . . . . 24 Flat lengths, Rough lengths . . . . . . . . . 25 Areas Angular areas . . . . . . . . . . . . . . . . . . . . . 26 Equilateral triangle, Polygons, Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Circular areas . . . . . . . . . . . . . . . . . . . . . 28 Volume and Surface area Cube, Cylinder, Pyramid . . . . . . . . . . . . 29 Truncated pyramid, Cone, Truncated cone, Sphere . . . . . . . . . . . . . 30 Composite solids . . . . . . . . . . . . . . . . . . 31 Mass General calculations . . . . . . . . . . . . . . . . 31 Linear mass density . . . . . . . . . . . . . . . . 31 Area mass density . . . . . . . . . . . . . . . . . 31 Centroids Centroids of lines . . . . . . . . . . . . . . . . . . 32 Centroids of plane areas . . . . . . . . . . . . 32
33
2.7
2.8
Bending, Torsion . . . . . . . . . . . . . . . . . . 47 Shape factors in strength . . . . . . . . . . . 48 Static moment, Section modulus, Moment of inertia . . . . . . . . . . . . . . . . . . 49 Comparison of various cross-sectional shapes . . . . . . . . . . . . . 50 Thermodynamics Temperatures, Linear expansion, Shrinkage . . . . . . . . . . . . . . 51 Quantity of heat . . . . . . . . . . . . . . . . . . . 51 Heat flux, Heat of combustion . . . . . . . 52 Electricity Ohm’s Law, Conductor resistance . . . . 53 Resistor circuits . . . . . . . . . . . . . . . . . . . 54 Types of current . . . . . . . . . . . . . . . . . . . 55 Electrical work and power . . . . . . . . . . . 56
5
Table of Contents
3 Technical drawing 3.1
3.2
3.3
3.4
3.5
Basic geometric constructions Lines and angles . . . . . . . . . . . . . . . . . . . 58 Tangents, Circular arcs, Polygons . . . . 59 Inscribed circles, Ellipses, Spirals . . . . . 60 Cycloids, Involute curves, Parabolas . . 61 Graphs Cartesian coordinate system . . . . . . . . 62 Graph types . . . . . . . . . . . . . . . . . . . . . . . 63 Drawing elements Fonts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Preferred numbers, Radii, Scales . . . . . 65 Drawing layout . . . . . . . . . . . . . . . . . . . . 66 Line types . . . . . . . . . . . . . . . . . . . . . . . . 67 Representation Projection methods . . . . . . . . . . . . . . . . 69 Views . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Sectional views . . . . . . . . . . . . . . . . . . . . 73 Hatching . . . . . . . . . . . . . . . . . . . . . . . . . 75 Entering dimensions Dimensioning rules . . . . . . . . . . . . . . . . 76 Diameters, Radii, Spheres, Chamfers, Inclines, Tapers, Arc dimensions . . . . . 78 Tolerance specifications . . . . . . . . . . . . 80 Types of dimensioning . . . . . . . . . . . . . 81 Simplified presentation in drawings . . 83
57 3.6
Machine elements Gear types . . . . . . . . . . . . . . . . . . . . . . . . 84 Roller bearings . . . . . . . . . . . . . . . . . . . . 85 Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Retaining rings, Springs . . . . . . . . . . . . 87 3.7 Workpiece elements Bosses, Workpiece edges . . . . . . . . . . . 88 Thread runouts, Thread undercuts . . . 89 Threads, Screw joints . . . . . . . . . . . . . . 90 Center holes, Knurls, Undercuts . . . . . . 91 3.8 Welding and Soldering Graphical symbols . . . . . . . . . . . . . . . . . 93 Dimensioning examples . . . . . . . . . . . . 95 3.9 Surfaces Hardness specifications in drawings . . 97 Form deviations, Roughness . . . . . . . . 98 Surface testing, Surface indications . . 99 3.10 ISO Tolerances and Fits Fundamentals . . . . . . . . . . . . . . . . . . . . 102 Basic hole and basic shaft systems . . 106 General Tolerances, Roller bearing fits . . . . . . . . . . . . . . . . . . . . . .110 Fit recommendations . . . . . . . . . . . . . .111 Geometric tolerancing . . . . . . . . . . . . .112 GD & T (Geometric Dimensioning & Tolerancing) . . . . . . .113
4 Materials science 4.1
4.2
4.3
4.4
4.5
4.6
Materials Material characteristics of solids . . . . 116 Material characteristics of liquids and gases . . . . . . . . . . . . . . . . . . . . . . . 117 Periodic table of the elements . . . . . . 118 Designation system for steels Definition and classification of steel . 120 Material codes, Designation . . . . . . . . 121 Steel types, Overview . . . . . . . . . . . 126 Structural steels . . . . . . . . . . . . . . . . . . 128 Case hardened, quenched and tempered, nitrided, free cutting steels . . . 132 Tool steels . . . . . . . . . . . . . . . . . . . . . . . 135 Stainless steels, Spring steels . . . . . . 136 Finished steel products Sheet, strip, pipes . . . . . . . . . . . . . . . . . 139 Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Heat treatment Iron-Carbon phase diagram . . . . . . . . 153 Processes . . . . . . . . . . . . . . . . . . . . . . . . 154 Cast iron materials Designation, Material codes . . . . . . . . 158 Classification . . . . . . . . . . . . . . . . . . . . . 159 Cast iron . . . . . . . . . . . . . . . . . . . . . . . . 160 Malleable cast iron, Cast steel . . . . . . 161
115 4.7
4.8
4.9
4.10
4.11
4.12
4.13 4.14
Foundry technology Patterns, Pattern equipment . . . . . . . . 162 Shrinkage allowances, Dimensional tolerances . . . . . . . . . . . . 163 Light alloys, Overview of Al alloys . . 164 Wrought aluminum alloys . . . . . . . . . 166 Aluminum casting alloys . . . . . . . . . . . 168 Aluminum profiles . . . . . . . . . . . . . . . . 169 Magnesium and titanium alloys . . . . . 172 Heavy non-ferrous metals, Overview . . . . . . . . . . . . . . . . . . . . . . . . 173 Designation system . . . . . . . . . . . . . . . 174 Copper alloys . . . . . . . . . . . . . . . . . . . . 175 Other metallic materials Composite materials, Ceramic materials . . . . . . . . . . . . . . . . 177 Sintered metals . . . . . . . . . . . . . . . . . . 178 Plastics, Overview . . . . . . . . . . . . . . 179 Thermoplastics . . . . . . . . . . . . . . . . . . . 182 Thermoset plastics, Elastomers . . . . . 184 Plastics processing . . . . . . . . . . . . . . . . 186 Material testing methods, Overview . . . . . . . . . . . . . . . . . . . . 188 Tensile testing . . . . . . . . . . . . . . . . . . . . 190 Hardness test . . . . . . . . . . . . . . . . . . . . 192 Corrosion, Corrosion protection . . 196 Hazardous materials . . . . . . . . . . . . 197
6
Table of Contents
5 Machine elements 5.1
5.2
5.3
5.4
5.5
5.6
Threads (overview) . . . . . . . . . . . . . 202 Metric ISO threads . . . . . . . . . . . . . . . . 204 Whitworth threads, Pipe threads . . . . 206 Trapezoidal and buttress threads . . . . 207 Thread tolerances . . . . . . . . . . . . . . . . . 208 Bolts and screws (overview) . . . . . 209 Designations, strength . . . . . . . . . . . . . 210 Hexagon head bolts & screws . . . . . . 212 Other bolts & screws . . . . . . . . . . . . . . 215 Screw joint calculations . . . . . . . . . . . . 221 Locking fasteners . . . . . . . . . . . . . . . . . 222 Widths across flats, Bolt and screw drive systems . . . . . . . . . . . . . . 223 Countersinks . . . . . . . . . . . . . . . . . . 224 Countersinks for countersunk head screws . . . . . . . . . . . . . . . . . . . . . 224 Counterbores for cap screws . . . . . . . 225 Nuts (overview) . . . . . . . . . . . . . . . . 226 Designations, Strength . . . . . . . . . . . . 227 Hexagon nuts . . . . . . . . . . . . . . . . . . . . 228 Other nuts . . . . . . . . . . . . . . . . . . . . . . . 231 Washers (overview) . . . . . . . . . . . . 233 Flat washers . . . . . . . . . . . . . . . . . . . . . 234 HV, Clevis pin, Conical spring washers . 235 Pins and clevis pins (overview) . . . 236 Dowel pins, Taper pins, Spring pins . 237
201 Grooved pins, Grooved drive studs, Clevis pins . . . . . . . . . . . . . . . . . . . . . . . 238 5.7 Shaft-hub connections Tapered and feather keys . . . . . . . . . . 239 Parallel and woodruff keys . . . . . . . . . 240 Splined shafts, Blind rivets . . . . . . . . . 241 Tool tapers . . . . . . . . . . . . . . . . . . . . . . . 242 5.8 Springs, components of jigs and tools Springs . . . . . . . . . . . . . . . . . . . . . . . . . 244 Drill bushings . . . . . . . . . . . . . . . . . . . . 247 Standard stamping parts . . . . . . . . . . . 251 5.9 Drive elements Belts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Transmission ratios . . . . . . . . . . . . . . . 259 Speed graph . . . . . . . . . . . . . . . . . . . . . 260 5.10 Bearings Plain bearings (overview) . . . . . . . . . . 261 Plain bearing bushings . . . . . . . . . . . . 262 Antifriction bearings (overview) . . . . . 263 Types of roller bearings . . . . . . . . . . . . 265 Retaining rings . . . . . . . . . . . . . . . . . . . 269 Sealing elements . . . . . . . . . . . . . . . . . 270 Lubricating oils . . . . . . . . . . . . . . . . . . . 271 Lubricating greases . . . . . . . . . . . . . . . 272
6 Production Engineering 6.1
6.2
6.3
6.4
6.5
Quality management Standards, Terminology . . . . . . . . . . . 274 Quality planning, Quality testing . . . . 276 Statistical analysis . . . . . . . . . . . . . . . . 277 Statistical process control . . . . . . . . . . 279 Process capability . . . . . . . . . . . . . . . . . 281 Production planning Time accounting according to REFA . 282 Cost accounting . . . . . . . . . . . . . . . . . . 284 Machine hourly rates . . . . . . . . . . . . . . 285 Machining processes Productive time . . . . . . . . . . . . . . . . . . 287 Machining coolants . . . . . . . . . . . . . . . 292 Cutting tool materials, Inserts, Tool holders . . . . . . . . . . . . . . . . . . . . . 294 Forces and power . . . . . . . . . . . . . . . . . 298 Cutting data: Drilling, Reaming, Turning . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Cutting data: Taper turning . . . . . . . . . 304 Cutting data: Milling . . . . . . . . . . . . . . . 305 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . 307 Cutting data: Grinding and honing . . 308 Material removal Cutting data . . . . . . . . . . . . . . . . . . . . . . 313 Processes . . . . . . . . . . . . . . . . . . . . . . . . 314 Separation by cutting Cutting forces . . . . . . . . . . . . . . . . . . . . 315
273 6.6
6.7
6.8
Shearing . . . . . . . . . . . . . . . . . . . . . . . . 316 Location of punch holder shank . . . . . 317 Forming Bending . . . . . . . . . . . . . . . . . . . . . . . . . 318 Deep drawing . . . . . . . . . . . . . . . . . . . . 320 Joining Welding processes . . . . . . . . . . . . . . . . 322 Weld preparation . . . . . . . . . . . . . . . . . 323 Gas welding . . . . . . . . . . . . . . . . . . . . . 324 Gas shielded metal arc welding . . . . . 325 Arc welding . . . . . . . . . . . . . . . . . . . . . . 327 Thermal cutting . . . . . . . . . . . . . . . . . . 329 Identification of gas cylinders . . . . . . . 331 Soldering and brazing . . . . . . . . . . . . . 333 Adhesive bonding . . . . . . . . . . . . . . . . 336 Workplace safety and environmental protection Prohibitive signs . . . . . . . . . . . . . . . . . . 338 Warning signs . . . . . . . . . . . . . . . . . . . . 339 Mandatory signs, Escape routes and rescue signs . . . . . 340 Information signs . . . . . . . . . . . . . . . . . 341 Danger symbols . . . . . . . . . . . . . . . . . . 342 Identification of pipe lines . . . . . . . . . . 343 Sound and noise . . . . . . . . . . . . . . . . . 344
7
Table of Contents
7 Automation and Information Technology 7.1
7.2
7.3
7.4
7.5
Basic terminology for control engineering Basic terminology, Code letters, Symbols . . . . . . . . . . . . . . . . . . . . . . . . 346 Analog controllers . . . . . . . . . . . . . . . . 348 Discontinuous and digital controllers . . 349 Binary logic . . . . . . . . . . . . . . . . . . . . . . 350 Electrical circuits Circuit symbols . . . . . . . . . . . . . . . . . . . 351 Designations in circuit diagrams . . . . 353 Circuit diagrams . . . . . . . . . . . . . . . . . . 354 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 355 Protective precautions . . . . . . . . . . . . . 356 Function charts and function diagrams Function charts . . . . . . . . . . . . . . . . . . . 358 Function diagrams . . . . . . . . . . . . . . . . 361 Pneumatics and hydraulics Circuit symbols . . . . . . . . . . . . . . . . . . . 363 Layout of circuit diagrams . . . . . . . . . 365 Controllers . . . . . . . . . . . . . . . . . . . . . . . 366 Hydraulic fluids . . . . . . . . . . . . . . . . . . . 368 Pneumatic cylinders . . . . . . . . . . . . . . . 369 Forces, Speeds, Power . . . . . . . . . . . . 370 Precision steel tube . . . . . . . . . . . . . . . 372 Programmable logic control PLC programming languages . . . . . . . 373 Ladder diagram (LD) . . . . . . . . . . . . . . 374 Function block language (FBL) . . . . . . 374
8 Material chart, Standards 8.1 8.2
7.6
7.7
7.8
345
Structured text (ST) . . . . . . . . . . . . . . . 374 Instruction list . . . . . . . . . . . . . . . . . . . 375 Simple functions . . . . . . . . . . . . . . . . . 376 Handling and robot systems Coordinate systems and axes . . . . . . . 378 Robot designs . . . . . . . . . . . . . . . . . . . . 379 Grippers, job safety . . . . . . . . . . . . . . . 380 Numerical Control (NC) technology Coordinate systems . . . . . . . . . . . . . . . 381 Program structure according to DIN . . 382 Tool offset and Cutter compensation . 383 Machining motions as per DIN . . . . . . . 384 Machining motions as per PAL (German association) . . . . . . . . . . . . . . 386 PAL programming system for turning . 388 PAL programming system for milling . 392 Information technology Numbering systems . . . . . . . . . . . . . . . 401 ASCII code . . . . . . . . . . . . . . . . . . . . . . . 402 Program flow chart, Structograms . . 403 WORD- and EXEL commands . . . . . . 405
407
International material comparison chart . . . . . . . . . . . . . . 407 DIN, DIN EN, ISO etc. standards . . 412
Subject index
417
8
Standards and other Regulations Standardization and Standards terms Standardization is the systematic achievement of uniformity of material and non-material objects, such as components, calculation methods, process flows and services for the benefit of the general public. Standards term
Example
Explanation
Standard
DIN 7157
A standard is the published result of standardization, e.g. the selection of certain fits in DIN 7157.
Part
DIN 30910-2
The part of a standard associated with other parts with the same main number. DIN 30910-2 for example describes sintered materials for filters, while Part 3 and 4 describe sintered materials for bearings and formed parts.
Supplement
DIN 743 Suppl. 1
A supplement contains information for a standard, however no additional specifications. The supplement DIN 743 Suppl. 1, for example, contains application examples of load capacity calculations for shafts and axles described in DIN 743.
Draft
E DIN 6316 (2007-02)
A draft standard contains the preliminary finished results of a standardization; this version of the intended standard is made available to the public for comments. For example, the planned new version of DIN 6316 for goose-neck clamps has been available to the public since February 2007 as Draft E DIN 6316.
Preliminary standard
DIN V 66304 (1991-12)
A preliminary standard contains the results of standardization which are not released by DIN as a standard, because of certain provisos. DIN V 66304, for example, discusses a format for exchange of standard part data for computer-aided design.
Issue date
DIN 76-1 (2004-06)
Date of publication which is made public in the DIN publication guide; this is the date at which time the standard becomes valid. DIN 76-1, which sets undercuts for metric ISO threads has been valid since June 2004 for example.
Types of Standards and Regulations (selection) Type International Standards (ISO standards) European Standards (EN standards)
Abbreviation
Explanation
Purpose and contents
ISO
International Organization for Standardization, Geneva (O and S are reversed in the abbreviation)
Simplifies the international exchange of goods and services, as well as cooperation in scientific, technical and economic areas.
EN
European Committee for Standardization (Comité Européen de Normalisation), Brussels
Technical harmonization and the associated reduction of trade barriers for the advancement of the European market and the coalescence of Europe. National standardization facilitates rationalization, quality assurance, environmental protection and common understanding in economics, technology, science, management and public relations.
DIN
DIN EN German Standards (DIN standards)
DIN ISO
German standard for which an international standard has been adopted without change.
DIN EN ISO
European standard for which an international standard has been adopted unchanged and the German version has the status of a German standard.
DIN VDE VDI Guidelines
VDI
VDE printed publications
VDE
DGQ publications
REFA sheets
Deutsches Institut für Normung e.V., Berlin (German Institute for Standardization) European standard for which the German version has attained the status of a German standard.
Printed publication of the VDE, which has the status of a German standard. Verein Deutscher Ingenieure e.V., These guidelines give an account of the curDüsseldorf (Society of German rent state of the art in specific subject areas Engineers) and contain, for example, concrete proceduVerband Deutscher Elektrotechniker ral guidelines for the performing calculations or designing processes in mechanical or e.V., Frankfurt (Organization of Gerelectrical engineering. man Electrical Engineers)
DGQ
Deutsche Gesellschaft für Qualität e.V., Recommendations in the area of quality technology. Frankfurt (German Association for Quality)
REFA
Association for Work Design/Work Structure, Industrial Organization and Corporate Development REFA e.V., Darmstadt
Recommendations in the area of production and work planning.
Table of Contents
9
1 Mathematics d 0 3
d
A=
p ·d 2
Numerical tables Square root, Area of a circle . . . . . . . . . . . . . . . . . . 10 Sine, Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Tangent, Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2
Trigonometric Functions Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sine, Cosine, Tangent, Cotangent . . . . . . . . . . . . . . Laws of sines and cosines . . . . . . . . . . . . . . . . . . . . Angles, Theorem of intersecting lines . . . . . . . . . .
13 13 14 14
P
Fundamentals of Mathematics Using brackets, powers, roots . . . . . . . . . . . . . . . . Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers of ten, Interest calculation . . . . . . . . . . . . . Percentage and proportion calculations . . . . . . . .
15 16 17 18
TD
4
1
1.0000
0.7854
2
1.4142
3.1416
3
1.7321
7.0686
=
opposite side hypotenuse
cosine
=
adjacent side hypotenuse
tangent
=
opposite side adjacent side
cotangent =
adjacent side opposite side
sine
1.1
1.3 3 x
5 + x
=
1 · (3 + 5) x
1.4 1 kW · h = 3.6 · 106 W · s
1.5
Symbols, Units Formula symbols, Mathematical symbols . . . . . . 19 SI quantities and units of measurement . . . . . . . . 20 Non-SI units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
M
MS
Lengths Calculations in a right triangle . . . . . . . . . . . . . . . . 23 Sub-dividing lengths, Arc length . . . . . . . . . . . . . . 24 Flat lengths, Rough lengths . . . . . . . . . . . . . . . . . . . 25
ME
m' in
kg m
1.6
Areas Angular areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Equilateral triangle, Polygons, Circle . . . . . . . . . . . 27 Circular areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.7
Volume and Surface area Cube, Cylinder, Pyramid . . . . . . . . . . . . . . . . . . . . . 29 Truncated pyramid, Cone, Truncated cone, Sphere 30 Composite solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.8
Mass General calculations . . . . . . . . . . . . . . . . . . . . . . . . . 31 Linear mass density . . . . . . . . . . . . . . . . . . . . . . . . . 31 Area mass density . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1 m
d
y S S1 xs
1.9
S2 s
y
x
Centroids Centroids of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Centroids of plane areas . . . . . . . . . . . . . . . . . . . . . 32
PE
A
S
10
Mathematics: 1.1 Numerical tables
Square root, Area of a circle M
P
TD
MS
ME
PE
A
S
d
0 3 d
A=
p ·d 2 4
d
0 3 d
A=
p ·d 2 4
d
0 3 d
A=
p ·d 2 4
d
0 3 d
A=
p ·d 2 4
1
1.0000
0.7854
51
7.1414
2042.82
101
10.049 9
8011.85
151
12.2882
17907.9
2
1.4142
3.1416
52
7.2111
2123.72
102
10.0995
8171.28
152
12.3288
18145.8
3
1.7321
7.0686
53
7.2801
2206.18
103
10.1489
8332.29
153
12.3693
18385.4
4
2.0000
12.5664
54
7.3485
2290.22
104
10.1980
8494.87
154
12.4097
18626.5
5
2.236 1
19.6350
55
7.4162
2375.83
105
10.2470
8659.01
155
12.4499
18869.2
6
2.4495
28.2743
56
7.4833
2463.01
106
10.2956
8824.73
156
12.4900
19113.4
7
2.6458
38.4845
57
7.5498
2551.76
107
10.3441
8992.02
157
12.5300
19359.3
8
2.8284
50.2655
58
7.6158
2642.08
108
10.3923
9160.88
158
12.5698
19606.7
9
3.0000
63.6173
59
7.6811
2733.97
109
10.4403
9331.32
159
12.6095
19855.7
10
3.1623
78.5398
60
7.746 0
2 827.43
110
10.488 1
9 503.32
160
12.649 1
20 106.2
11
3.3166
95.0332
61
7.8102
2922.47
111
10.5357
9676.89
161
12.6886
20358.3
12
3.4641
113.097
62
7.8740
3019.07
112
10.5830
9852.03
162
12.7279
20612.0
13
3.6056
132.732
63
7.9373
3117.25
113
10.6301
10028.7
163
12.7671
20867.2
14
3.7417
153.938
64
8.0000
3216.99
114
10.6771
10207.0
164
12.8062
21124.1
15
3.8730
176.715
65
8.0623
3318.31
115
10.7238
10386.9
165
12.8452
21382.5
16
4.0000
201.062
66
8.1240
3421.19
116
10.7703
10568.3
166
12.8841
21642.4
17
4.1231
226.980
67
8.1854
3525.65
117
10.8167
10751.3
167
12.9228
21904.0
18
4.2426
254.469
68
8.2462
3631.68
118
10.8628
10935.9
168
12.9615
22167.1
19
4.3589
283.529
69
8.3066
3739.28
119
10.9087
11122.0
169
13.0000
22431.8
20
4.472 1
314.159
70
8.366 6
3 848.45
120
10.9545
11309.7
170
13.038 4
22 698.0
21
4.5826
346.361
71
8.4261
3959.19
121
11.0000
11499.0
171
13.0767
22965.8
22
4.6904
380.133
72
8.4853
4071.50
122
11.0454
11689.9
172
13.1149
23235.2
23
4.7958
415.476
73
8.5440
4185.39
123
11.0905
11882.3
173
13.1529
23506.2
24
4.8990
452.389
74
8.6023
4300.84
124
11.1355
12076.3
174
13.1909
23778.7
25
5.0000
490.874
75
8.6603
4417.86
125
11.1803
12271.8
175
13.2288
24052.8
26
5.0990
530.929
76
8.7178
4536.46
126
11.2250
12469.0
176
13.2665
24328.5
27
5.1962
572.555
77
8.7750
4656.63
127
11.2694
12667.7
177
13.3041
24605.7
28
5.2915
615.752
78
8.8318
4778.36
128
11.3137
12868.0
178
13.3417
24884.6
29
5.3852
660.520
79
8.8882
4901.67
129
11.3578
13069.8
179
13.3791
25164.9
30
5.477 2
706.858
80
8.944 3
5 026.55
130
11.4018
13273.2
180
13.416 4
25 446.9
31
5.5678
754.768
81
9.0000
5153.00
131
11.4455
13478.2
181
13.4536
25730.4
32
5.6569
804.248
82
9.0554
5281.02
132
11.4891
13684.8
182
13.4907
26015.5
33
5.7446
855.299
83
9.1104
5410.61
133
11.5326
13892.9
183
13.5277
26302.2
34
5.8310
907.920
84
9.1652
5541.77
134
11.5758
14102.6
184
13.5647
26590.4
35
5.9161
962.113
85
9.2195
5674.50
135
11.6190
14313.9
185
13.6015
26880.3
36
6.0000
1017.88
86
9.2736
5808.80
136
11.6619
14526.7
186
13.6382
27171.6
37
6.0828
1075.21
87
9.3274
5944.68
137
11.7047
14741.1
187
13.6748
27464.6
38
6.1644
1134.11
88
9.3808
6082.12
138
11.7473
14957.1
188
13.7113
27759.1
39
6.2450
1194.59
89
9.4340
6221.14
139
11.7898
15174.7
189
13.7477
28055.2
40
6.324 6
1256.64
90
9.486 8
6 361.73
140
11.8322
15393.8
190
13.784 0
28 352.9
41
6.4031
1320.25
91
9.5394
6503.88
141
11.8743
15614.5
191
13.8203
28652.1
42
6.4807
1385.44
92
9.5917
6647.61
142
11.9164
15836.8
192
13.8564
28952.9
43
6.5574
1452.20
93
9.6437
6792.91
143
11.9583
16060.6
193
13.8924
29255.3
44
6.6332
1520.53
94
9.6954
6939.78
144
12.0000
16286.0
194
13.9284
29559.2
45
6.7082
1590.43
95
9.7468
7088.22
145
12.0416
16513.0
195
13.9642
29864.8
46
6.7823
1661.90
96
9.7980
7238.23
146
12.0830
16741.5
196
14.0000
30171.9
47
6.8557
1734.94
97
9.8489
7389.81
147
12.1244
16971.7
197
14.0357
30480.5
48
6.9282
1809.56
98
9.8995
7542.96
148
12.1655
17203.4
198
14.0712
30790.7
49
7.0000
1885.74
99
9.9499
7697.69
149
12.2066
17436.6
199
14.1067
31102.6
50
7.071 1
1963.50
100
10.000 0
7 853.98
150
12.2474
17671.5
200
14.142 1
31 415.9
Table values of 0 3 d and A are rounded off.
11
Mathematics: 1.1 Numerical tables
Values of Sine and Cosine Trigonometric Functions sine 0° to 45°
degrees
º
sine 45° to 90°
degrees
minutes
º
M
minutes
0*
15 *
30 *
45 *
60 *
0° 1° 2° 3° 4°
0.0000 0.0175 0.0349 0.0523 0.0698
0.0044 0.0218 0.0393 0.0567 0.0741
0.0087 0.0262 0.0436 0.0610 0.0785
0.0131 0.0305 0.0480 0.0654 0.0828
0.0175 0.0349 0.0523 0.0698 0.0872
89° 88° 87° 86° 85°
45° 46° 47° 48° 49°
0.7071 0.7193 0.7314 0.7431 0.7547
0.7102 0.7224 0.7343 0.7461 0.7576
0.7133 0.7254 0.7373 0.7490 0.7604
0.7163 0.7284 0.7402 0.7518 0.7632
0.7193 0.7314 0.7431 0.7547 0.7660
44° 43° 42° 41° 40°
5° 6° 7° 8° 9°
0.0872 0.1045 0.1219 0.1392 0.1564
0.0915 0.1089 0.1262 0.1435 0.1607
0.0958 0.1132 0.1305 0.1478 0.1650
0.1002 0.1175 0.1349 0.1521 0.1693
0.1045 0.1219 0.1392 0.1564 0.1736
84° 83° 82° 81° 80°
50° 51° 52° 53° 54°
0.7660 0.7771 0.7880 0.7986 0.8090
0.7688 0.7799 0.7907 0.8013 0.8116
0.7716 0.7826 0.7934 0.8039 0.8141
0.7744 0.7853 0.7960 0.8064 0.8166
0.7771 0.7880 0.7986 0.8090 0.8192
39° 38° 37° 36° 35°
10° 11° 12° 13° 14°
0.1736 0.1908 0.2079 0.2250 0.2419
0.1779 0.1951 0.2122 0.2292 0.2462
0.1822 0.1994 0.2164 0.2334 0.2504
0.1865 0.2036 0.2207 0.2377 0.2546
0.1908 0.2079 0.2250 0.2419 0.2588
79° 78° 77° 76° 75°
55° 56° 57° 58° 59°
0.8192 0.8290 0.8387 0.8480 0.8572
0.8216 0.8315 0.8410 0.8504 0.8594
0.8241 0.8339 0.8434 0.8526 0.8616
0.8266 0.8363 0.8457 0.8549 0.8638
0.8290 0.8387 0.8480 0.8572 0.8660
34° 33° 32° 31° 30°
15° 16° 17° 18° 19°
0.2588 0.2756 0.2924 0.3090 0.3256
0.2630 0.2798 0.2965 0.3132 0.3297
0.2672 0.2840 0.3007 0.3173 0.3338
0.2714 0.2882 0.3049 0.3214 0.3379
0.2756 0.2924 0.3090 0.3256 0.3420
74° 73° 72° 71° 70°
60° 61° 62° 63° 64°
0.8660 0.8746 0.8829 0.8910 0.8988
0.8682 0.8767 0.8850 0.8930 0.9007
0.8704 0.8788 0.8870 0.8949 0.9026
0.8725 0.8809 0.8890 0.8969 0.9045
0.8746 0.8829 0.8910 0.8988 0.9063
29° 28° 27° 26° 25°
20° 21° 22° 23° 24°
0.3420 0.3584 0.3746 0.3907 0.4067
0.3461 0.3624 0.3786 0.3947 0.4107
0.3502 0.3665 0.3827 0.3987 0.4147
0.3543 0.3706 0.3867 0.4027 0.4187
0.3584 0.3746 0.3907 0.4067 0.4226
69° 68° 67° 66° 65°
65° 66° 67° 68° 69°
0.9063 0.9135 0.9205 0.9272 0.9336
0.9081 0.9153 0.9222 0.9288 0.9351
0.9100 0.9171 0.9239 0.9304 0.9367
0.9118 0.9188 0.9255 0.9320 0.9382
0.9135 0.9205 0.9272 0.9336 0.9397
24° 23° 22° 21° 20°
25° 26° 27° 28° 29°
0.4226 0.4384 0.4540 0.4695 0.4848
0.4266 0.4423 0.4579 0.4733 0.4886
0.4305 0.4462 0.4617 0.4772 0.4924
0.4344 0.4501 0.4656 0.4810 0.4962
0.4384 0.4540 0.4695 0.4848 0.5000
64° 63° 62° 61° 60°
70° 71° 72° 73° 74°
0.9397 0.9455 0.9511 0.9563 0.9613
0.9412 0.9469 0.9524 0.9576 0.9625
0.9426 0.9483 0.9537 0.9588 0.9636
0.9441 0.9497 0.9550 0.9600 0.9648
0.9455 0.9511 0.9563 0.9613 0.9659
19° 18° 17° 16° 15°
30° 31° 32° 33° 34°
0.5000 0.5150 0.5299 0.5446 0.5592
0.5038 0.5188 0.5336 0.5483 0.5628
0.5075 0.5225 0.5373 0.5519 0.5664
0.5113 0.5262 0.5410 0.5556 0.5700
0.5150 0.5299 0.5446 0.5592 0.5736
59° 58° 57° 56° 55°
75° 76° 77° 78° 79°
0.9659 0.9703 0.9744 0.9781 0.9816
0.9670 0.9713 0.9753 0.9790 0.9825
0.9681 0.9724 0.9763 0.9799 0.9833
0.9692 0.9734 0.9772 0.9808 0.9840
0.9703 0.9744 0.9781 0.9816 0.9848
14° 13° 12° 11° 10°
35° 36° 37° 38° 39°
0.5736 0.5878 0.6018 0.6157 0.6293
0.5771 0.5913 0.6053 0.6191 0.6327
0.5807 0.5948 0.6088 0.6225 0.6361
0.5842 0.5983 0.6122 0.6259 0.6394
0.5878 0.6018 0.6157 0.6293 0.6428
54° 53° 52° 51° 50°
80° 81° 82° 83° 84°
0.9848 0.9877 0.9903 0.9925 0.9945
0.9856 0.9884 0.9909 0.9931 0.9950
0.9863 0.9890 0.9914 0.9936 0.9954
0.9870 0.9897 0.9920 0.9941 0.9958
0.9877 0.9903 0.9925 0.9945 0.9962
9° 8° 7° 6° 5°
40° 41° 42° 43° 44°
0.6428 0.6561 0.6691 0.6820 0.6947
0.6461 0.6593 0.6724 0.6852 0.6978
0.6494 0.6626 0.6756 0.6884 0.7009
0.6528 0.6659 0.6788 0.6915 0.7040
0.6561 0.6691 0.6820 0.6947 0.7071
49° 48° 47° 46° 45°
85° 86° 87° 88° 89°
0.9962 0.9976 0.9986 0.9994 0.99985
0.9966 0.9979 0.9988 0.9995 0.99991
0.9969 0.9981 0.9990 0.9997 0.99996
0.9973 0.9984 0.9992 0.9998 0.99999
0.9976 0.9986 0.9994 0.99985 1.0000
4° 3° 2° 1° 0°
60*
45 *
30 *
15*
0*
60*
45 *
30 *
15 *
0*
minutes cosine 45° to 90°
degrees
0*
15 *
30 *
45 *
minutes cosine 0° to 45°
Table values of the trigonometric functions are rounded off to four decimal places.
60 *
degrees
P
TD
MS
ME
PE
A
S
12
Mathematics: 1.1 Numerical tables
Values of Tangent and Cotangent Trigonometric Functions M
º
P
TD
MS
ME
PE
A
tangent 0° to 45°
degrees
minutes
º
minutes
0*
15 *
30 *
45 *
60 *
0° 1° 2° 3° 4°
0.0000 0.0175 0.0349 0.0524 0.0699
0.0044 0.0218 0.0393 0.0568 0.0743
0.0087 0.0262 0.0437 0.0612 0.0787
0.0131 0.0306 0.0480 0.0655 0.0831
0.0175 0.0349 0.0524 0.0699 0.0875
89° 88° 87° 86° 85°
45° 46° 47° 48° 49°
1.0000 1.0355 1.0724 1.1106 1.1504
1.0088 1.0446 1.0818 1.1204 1.1606
1.0176 1.0538 1.0913 1.1303 1.1708
1.0265 1.0630 1.1009 1.1403 1.1812
1.0355 1.0724 1.1106 1.1504 1.1918
44° 43° 42° 41° 40°
5° 6° 7° 8° 9°
0.0875 0.1051 0.1228 0.1405 0.1584
0.0919 0.1095 0.1272 0.1450 0.1629
0.0963 0.1139 0.1317 0.1495 0.1673
0.1007 0.1184 0.1361 0.1539 0.1718
0.1051 0.1228 0.1405 0.1584 0.1763
84° 83° 82° 81° 80°
50° 51° 52° 53° 54°
1.1918 1.2349 1.2799 1.3270 1.3764
1.2024 1.2460 1.2915 1.3392 1.3891
1.2131 1.2572 1.3032 1.3514 1.4019
1.2239 1.2685 1.3151 1.3638 1.4150
1.2349 1.2799 1.3270 1.3764 1.4281
39° 38° 37° 36° 35°
10° 11° 12° 13° 14°
0.1763 0.1944 0.2126 0.2309 0.2493
0.1808 0.1989 0.2171 0.2355 0.2540
0.1853 0.2035 0.2217 0.2401 0.2586
0.1899 0.2080 0.2263 0.2447 0.2633
0.1944 0.2126 0.2309 0.2493 0.2679
79° 78° 77° 76° 75°
55° 56° 57° 58° 59°
1.4281 1.4826 1.5399 1.6003 1.6643
1.4415 1.4966 1.5547 1.6160 1.6808
1.4550 1.5108 1.5697 1.6319 1.6977
1.4687 1.5253 1.5849 1.6479 1.7147
1.4826 1.5399 1.6003 1.6643 1.7321
34° 33° 32° 31° 30°
15° 16° 17° 18° 19°
0.2679 0.2867 0.3057 0.3249 0.3443
0.2726 0.2915 0.3105 0.3298 0.3492
0.2773 0.2962 0.3153 0.3346 0.3541
0.2820 0.3010 0.3201 0.3395 0.3590
0.2867 0.3057 0.3249 0.3443 0.3640
74° 73° 72° 71° 70°
60° 61° 62° 63° 64°
1.7321 1.8040 1.8807 1.9626 2.0503
1.7496 1.8228 1.9007 1.9840 2.0732
1.7675 1.8418 1.9210 2.0057 2.0965
1.7856 1.8611 1.9416 2.0278 2.1203
1.8040 1.8807 1.9626 2.0503 2.1445
29° 28° 27° 26° 25°
20° 21° 22° 23° 24°
0.3640 0.3839 0.4040 0.4245 0.4452
0.3689 0.3889 0.4091 0.4296 0.4505
0.3739 0.3939 0.4142 0.4348 0.4557
0.3789 0.3990 0.4193 0.4400 0.4610
0.3839 0.4040 0.4245 0.4452 0.4663
69° 68° 67° 66° 65°
65° 66° 67° 68° 69°
2.1445 2.2460 2.3559 2.4751 2.6051
2.1692 2.2727 2.3847 2.5065 2.6395
2.1943 2.2998 2.4142 2.5386 2.6746
2.2199 2.3276 2.4443 2.5715 2.7106
2.2460 2.3559 2.4751 2.6051 2.7475
24° 23° 22° 21° 20°
25° 26° 27° 28° 29°
0.4663 0.4877 0.5095 0.5317 0.5543
0.4716 0.4931 0.5150 0.5373 0.5600
0.4770 0.4986 0.5206 0.5430 0.5658
0.4823 0.5040 0.5261 0.5486 0.5715
0.4877 0.5095 0.5317 0.5543 0.5774
64° 63° 62° 61° 60°
70° 71° 72° 73° 74°
2.7475 2.9042 3.0777 3.2709 3.4874
2.7852 2.9459 3.1240 3.3226 3.5457
2.8239 2.9887 3.1716 3.3759 3.6059
2.8636 3.0326 3.2205 3.4308 3.6680
2.9042 3.0777 3.2709 3.4874 3.7321
19° 18° 17° 16° 15°
30° 31° 32° 33° 34°
0.5774 0.6009 0.6249 0.6494 0.6745
0.5832 0.6068 0.6310 0.6556 0.6809
0.5890 0.6128 0.6371 0.6619 0.6873
0.5949 0.6188 0.6432 0.6682 0.6937
0.6009 0.6249 0.6494 0.6745 0.7002
59° 58° 57° 56° 55°
75° 76° 77° 78° 79°
3.7321 4.0108 4.3315 4.7046 5.1446
3.7983 4.0876 4.4194 4.8077 5.2672
3.8667 4.1653 4.5107 4.9152 5.3955
3.9375 4.2468 4.6057 5.0273 5.5301
4.0108 4.3315 4.7046 5.1446 5.6713
14° 13° 12° 11° 10°
35° 36° 37° 38° 39°
0.7002 0.7265 0.7536 0.7813 0.8098
0.7067 0.7332 0.7604 0.7883 0.8170
0.7133 0.7400 0.7673 0.7954 0.8243
0.7199 0.7467 0.7743 0.8026 0.8317
0.7265 0.7536 0.7813 0.8098 0.8391
54° 53° 52° 51° 50°
80° 81° 82° 83° 84°
5.6713 6.3138 7.1154 8.1443 9.5144
5.8197 5.9758 6.1402 6.4971 6.6912 6.8969 7.3479 7.5958 7.8606 8.4490 8.7769 9.1309 9.9310 10.3854 10.8829
6.3138 7.1154 8.1443 9.5144 11.4301
9° 8° 7° 6° 5°
40° 41° 42° 43° 44°
0.8391 0.8693 0.9004 0.9325 0.9657
0.8466 0.8770 0.9083 0.9407 0.9742
0.8541 0.8847 0.9163 0.9490 0.9827
0.8617 0.8925 0.9244 0.9573 0.9913
0.8693 0.9004 0.9325 0.9657 1.0000
49° 48° 47° 46° 45°
85° 86° 87° 88° 89°
11.4301 14.3007 19.0811 28.6363 57.2900
12.0346 15.2571 20.8188 32.7303 76.3900
12.7062 16.3499 22.9038 38.1885 114.5887
13.4566 17.6106 25.4517 45.8294 229.1817
14.3007 19.0811 28.6363 57.2900
4° 3° 2° 1° 0°
60 *
45 *
30 *
15 *
0*
60 *
S
tangent 45° to 90°
degrees
45 *
30 *
15 * minutes
cotangent 45° to 90°
0*
º degrees
0*
15 *
30 *
45 *
minutes cotangent 0° to 45°
Table values of the trigonometric functions are rounded off to four decimal places.
60 *
6
º degrees
13
Mathematics: 1.2 Trigonometric Functions
Trigonometric functions of right triangles Definitions
M
Designations in a right triangle
c hypotenuse
a opposite side of å
å
¿
a adjacent side of ¿
b opposite side of ¿
for @ a
for @ b
sine
=
opposite side hypotenuse
sin a =
a c
sin b =
b c
cosine
=
adjacent side hypotenuse
cos a =
b c
cos b =
a c
=
opposite side adjacent side
tan a
cotangent =
adjacent side opposite side
cot a =
b adjacent side of å c hypotenuse
Application
Definitions of the ratios of the sides
tangent
a = b
tan b
b a
P
b = a
cot b =
a b
Graph of the trigonometric functions between 0° and 360° Representation on a unit circle
¡¡
90} +
cot ¿(-) ) + (
å
n i s
-
) + (
¿
¿
180}
cot å(+)
å
n i s
cos å(+)
cos ¿(-) 1 =
Graph of the trigonometric functions
¡¡
¡ +1 ) + (
å
n a t
+ 0} 360}
) (
¿
r
n a t
e u l a v
¡|
s
å
i n
c
o t
n a
å
å
t
n o i t c n u f
0}
å
o t
n a
å
t
90}
å
s c o
c
180}
¡
¡|
360} å
270}
-1
270}
¡¡¡
TD
MS
å
¡¡¡
n
a t
The values of the trigonometric functions of angles > 90° can be derived from the values of the angles between 0° and 90° and then read from the tables (pages 11 and 12). Refer to the graphed curves of the trigonometric functions for the correct sign. Calculators with trigonometric functions display both the value and sign for the desired angle.
ME
Example: Relationships for Quadrant II Relationships
Example: Function values for the angle 120° ( a = 30° in the formulae)
sin (90° + a) = +cos
sin (90° + 30°) = sin 120° = +0.8660
cos 30° = +0.8660
cos (90° + 30°) = cos 120° = –0.5000
–sin 30° = –0.5000
tan (90° + 30°) = tan 120° = –1.7321
–cot 30° = –1.7321
a cos (90° + a) = –sin a tan (90° + a) = –cot a
PE
Function values for selected angles Function
0°
90°
180°
270°
360°
Function
0°
90°
180°
270°
360°
sin
0
+1
0
–1
0
tan
0
6
0
6
0
cos
+1
0
–1
0
+1
cot
6
0
6
0
6
Relationships between the functions of an angle sin2 a + cos2 a = 1
A
tan a · cot a = 1
1 å
cos å
sin å
tan a =
sin a cos a
cot a =
cos a sin a
Example: Calculation of tan a from sin a and cos a for a = 30°: tan a = sin a /cos a = 0.5000/ 0.8660 = 0.5774
S
14
Mathematics: 1.2 Trigonometric Functions
Trigonometric functions of oblique triangles, Angles, Theorem of intersecting lines M
Law of sines and Law of cosines
©
b
a
Law of sines
Law of cosines
a : b : c = sin a : sin b : sin g
a 2 = b 2 + c 2 – 2 · b · c · cos a b 2 = a 2 + c 2 – 2 · a · c · cos b
¿
å
a
c
sinα
=
b sin β
=
c
c 2 = a 2 + b 2 – 2 · a · b · cos g
sin γ
P Application in calculating sides and angles Calculation of sides using the Law of sines
TD
a = b = c =
b ·sin α sin β
a · sin β sinα
a · sin γ sinα
=
=
using the Law of cosines
c · sinα
=
Calculation of angles
b 2 + c 2 – 2 · b · c · cos α
sinα =
b = a 2 + c 2 – 2 · a · c · cos β
sin β =
a
sinγ
c ·sin β sinγ
=
b ·sin γ c
sin β
using the Law of sines
=
a 2 + b 2 – 2 · a · b · cos γ
sinγ
=
a · sin β b b · sinα a c · sinα a
=
=
=
a · sin γ
using the Law of cosines
cos α
=
cos β
=
cos γ
=
b 2 + c 2 – a 2 2 · b · c
c b · sin γ c c · sin β b
a 2 + c 2 – b 2 2 · a · c
a 2 + b 2 – c 2 2 · a · b
Types of angles
MS
Corresponding angles
¿ g2
If two parallels g 1 and g 2 are intersected by a straight line g , there are geometrical interrelationships between the corresponding, opposite, alternate and adjacent angles.
a=b Opposite angles
b=d
¶ å
ME
Alternate angles ©
a=d
g1
Adjacent angles
g
a + g = 180°
Sum of angles in a triangle
PE
Sum of angles in a triangle
©
b
a
In every triangle the sum of the interior angles equals 180°.
a + b + g = 180°
¿
å
c
A
Theorem of intersecting lines
c 1
C
c
C1 1
å
A
S
b b1
a
B
B1
a
If two lines extending from Point A are intersected by two parallel lines BC and B1C1, the segments of the parallel lines and the corresponding ray segments of the lines extending from A form equal ratios.
Theorem of intersecting lines
a a 1 a b
=
=
b b 1
=
c c 1
a 1
b
b 1
c
=
b 1 c 1
15
Mathematics: 1.3 Fundamentals
Using brackets, powers and roots Calculations with brackets
M
Type
Explanation
Example
Factoring out
Common factors (divisors) in addition and subtraction are placed before a bracket.
3·x
+5·x = x
3
5
+
x
Expanding bracketed terms
Binomial formulae
1
=
x
x
· (3 + 5) = 8 · x
· (3 + 5)
A fraction bar combines terms in the same manner as brackets.
a + b
A bracketed term is multiplied by a value (number, variable, another bracketed term), by multiplying each term inside the brackets by this value.
5 · (b + c ) = 5b + 5c
A bracketed term is divided by a value (number, variable, another bracketed term), by dividing each term inside the bracket by this value.
(a + b) : c
A binomial formula is a formula in which the term ( a + b ) or (a – b ) is multiplied by itself.
(a + b )2 = a 2 + 2ab + b 2
2
· h = (a + b ) ·
h
2
P
(a + b ) · (c – d ) = ac – ad + bc – bd
a −b
=
5
= a :c + b :c
a
5
–
b
5
TD
(a – b )2 = a 2 – 2ab + b 2 (a + b ) · (a – b ) = a 2 – b 2
Multiplication/division and addition/subtraction calculations
In mixed equations, the bracketed terms must be solved first. Then multiplication and division calculations are performed, and finally addition and subtraction.
a · (3x – 5x ) – b · (12y – 2y )
a base;
a x = y
= a · (–2x ) – b · 10y = –2 ax – 10by
Powers Definitions
x exponent;
y exponential value
Product of identical factors
MS
a · a · a · a =
a 4
4 · 4 · 4 · 4 = 4 4 = 256 Addition Subtraction
Powers with the same base and the same exponents are treated like equal numbers.
3 a 3 + 5 a 3 – 4 a 3
Multiplication Division
Powers with the same base are multiplied (divided) by adding (subtracting) the exponents and keeping the base.
a4 · a 2 = a · a · a · a · a · a = a 6
= a 3 · (3 + 5 – 4) = 4 a 3 24 · 22 = 2(4+2) = 26 = 64 32 ÷ 33 = 3(2–3) = 3–1 = 1/3
Negative exponent
Numbers with negative exponents can also be written as fractions. The base is then given a positive exponent and is placed in the denominator.
m −1 = a −3 =
1 m1
=
ME
1 m
1 a 3
4
Fractions in exponents
Powers with fractional exponents can also be written as roots.
a3 =
Zero in exponents
Every power with a zero exponent has the value of one.
(m + n )0 = 1
3
a 4
PE
a 4 ÷ a 4 = a (4–4) = a 0 = 1
20 = 1
Roots Definitions
x root’s exponent;
Signs
Even number exponents of the root give positive and negative values, if the radicand is positive. A negative radicand results in an imaginary number.
2
9
2
−9 = +
Odd number exponents of the root give positive values if the radicand is positive and negative values if the radicand is negative.
3
8
3
−8 = −2
a radicand;
y root value
Addition Subtraction
Identical root expressions can be added and subtracted.
Multiplication Division
Roots with the same exponents are multiplied (divided) by taking the root of the product (quotient) of the radicands.
x
a
=
y or a1/ x
= ±
=
y
3
A
3i
=2
a +3
a −2
n
a· b =
n
3
a
3
n
=3
a n
n
a =2
ab
a
S
16
Mathematics: 1.3 Fundamentals
Types of equations, Rules of transformation M
Equations Type
Explanation
Example
Variable equation
Equivalent terms (formula terms of equal value ) form r elationships between variables (see also, Rules of transformation).
v = p · d · n
Compatible units equation
Immediate conversion of units and constants to an SI unit in the result.
(a + b )2 = a 2 + 2ab + b 2 P
M ·n
=
Only used in special cases, e.g. if engineering parameters are specified or for simplification.
P
; P in kW, if 9550 n in 1/min and M in Nm
Single variable equation
Calculation of the value of a variable.
x + 3= 8 x = 8 – 3 = 5
Function equation
Assigned function equation: y is a function of x with x as the independent variable; y as the dependent variable.
y = f (x ) R
real numbers
ª
The number pair (x ,y ) of a value table form the graph of the function in the ( x ,y ) coordinate system.
TD
Constant function
y = f (x ) = b
The graph is a line parallel to the x -axis.
MS
Proportional function
y = f (x ) = mx
The graph is a straight line through the origin.
y = 2x
Linear function
y = f (x ) = mx + b
The graph is a straight line with slope m and y intercept b (example below).
y = 0.5 x + 1
Quadratic function
y = f (x ) = x 2
Every quadratic (example below). linear function y = mx + b
3
function
example: y =0.5 x + 1
ME
as
a
parabola
y = a 2x 2 + a 1x + a 0
quadratic function y = x 2
2 y
graphs
3 2 y
m =0.5
1
example: y =0.5· x 2
1
b = 1
2 1
1 1
2
3
2 1
1 1
x
2
3
x
Rules of transformation Equations are usually transformed to obtain an equation in which the unknown variable stands alone on the left side of the equation.
PE
A
Addition Subtraction
Multiplication Division
æ– 5
In the equations x + 5 = 15 and x + 5 – 5 = 15 – 5, x has the same value, i. e. the equations are equivalent.
x + 5 = 15 x + 5 – 5 = 15 – 5 x = 10 y –c =d y – c +c = d + c y = d + c
It is possible to multiply or divide each side of the equation by the same number.
a·x = b a · x b
æ÷ a
The same number can be added or subtracted from both sides.
=
a x
Powers
The expressions on both sides of the equations can be raised to the same exponential power.
=
x ( x )2
x
S
Roots
The root of the expressions on both sides of the equation can be taken using the same root exponent.
x2 ( x )2
x
æ+ c
a b a
= a + b
æ ()2
= (a + b )2 =
a2 + 2 ab + b 2
= a + b = = ±
a + b a + b
æ
17
Mathematics: 1.3 Fundamentals
Decimal multiples and factors of units, Interest calculation Decimal multiples and factors of units
cf. DIN 1301-1 (2002-10)
Mathematics Power of ten
Name
SI units Prefix
Multiplication factor
Examples
Name
Character
Unit
Meaning
1018 1015 1012 109 106 103 102 101 100
quintillion quadrillion trillion billion million thousand hundred ten one
1 000 000 000 000 000 000 1 000 000 000 000 000 1 000 000 000 000 1 000 000 000 1 000 000 1 000 100 10 1
exa peta tera giga mega kilo hecto deca –
E P T G M k h da –
Em Pm TV GW MW kN hl dam m
10 18 10 15 10 12 10 9 10 6 10 3 10 2 10 1 100
10–1 10–2 10–3 10–6 10–9 10–12 10–15 10–18
tenth hundredth thousandth millionth billionth trillionth quadrillionth quintillionth
0.1 0.01 0.001 0.000 001 0.000 000 001 0.000 000 000 001 0.000 000 000 000 001 0.000 000 000 000 000 001
deci centi milli micro nano pico femto atto
d c m
dm cm mV mA nm pF fF am
10 -1 meters 10 -2 meters 10 -3 volts 10–6 ampere 10 -9 meters 10 –12 farad 10 -15 farads 10 -18 meters
values <1
10
3
10
2
10
1
1 10
10
1
n p f a
10
2
10
0.07 =
3
I interest r interest rate per year
t
Interest
time in days, interest period
I
1st example:
I
=
% 6 ; a
$ 2800.00; r = $ 2800.00 · 6
=
% a
t
=
I
=
I
=
P·r =
· t
100% · 360
? 1 interest year (1a) = 360 days (360 d)
· 0.5 a =
100%
1/ a; 2
$ 84. 00
360 d = 12 months 1 interest month = 30 days
2nd example: P
$ 4800.00; r
=
$ 4800.00· 5.1 =
%
5.1 ; t a % a
100% · 360
=
·50d d a
=
50 d; I
=
?
principle amount accumulated
PE
$ 34. 00
Compound interest calculation for one-time payment P A
MS
ME
principle amount accumulated
P
TD
7 = 7 · 10–2 100
Simple interest P A
P
Examples: 4300 = 4.3 · 1000 = 4.3 · 10 3 14638 = 1.4638 · 10 4
10 100 1000
0
m
meters meters volts watts watts newtons liters meters meter
Numbers greater than 1 are expressed with positive exponents and numbers less than 1 are expressed with negative exponents.
>1
1 1 1 1000 100 10
M
I interest r interest rate per year
n q
time compounding factor
A Amount accumulated
A = P · q n
Example:
$ 8000.00; n = 7 years; r 6.5% = 1.065 q = 1 + 100%
P
A
=
=
P · q n
= =
=
6.5 %; A = ?
$ 8000.00 · 1.0657 = $ 8000.00 · 1. 553986 $ 12431. 89
Compounding factor
q = 1 +
r
100%
S
18
Mathematics: 1.3 Fundamentals
Percentage calculation, Proportion calculations M
Percentage calculation Percent value
The percentage rate gives the fraction of the base value in hundredths. The base value is the value from which the percentage is to be calculated. The percent value is the amount representing the percentage of the base value. P r percentage rate, in percent
P v percent value
P v
B v base value.
1st example:
P
P v
=
100 %
B v · P r
100 %
Percentage rate
Workpiece rough part weight 250 kg (base value); material loss 2 % (percentage rate); material loss in kg = ? (percent value) B v · P r
=
=
P r
250 kg · 2 % = 5 kg 100 %
=
P v B v
· 100%
2nd example: Rough weight of a casting 150 kg; weight after machining 126 kg; weight percent rate ( %) of material loss?
TD
P r
=
P v B v
· 100% =
150 kg–126 kg 150 kg
· 100% = 16%
Proportion calculations Three steps for calculating direct proportional ratios Example:
MS
80
60 elbow pipes weigh 330 kg. What is the weight of 35 elbow pipes? 1st step:
60
2nd step:
40
s t i n u
20 0
Known data
60 elbow pipes weigh 330 kg.
Calculate the unit weight by dividing
1 elbow pipe weighs
0
100 200 kg 300 weight
3rd step:
Calculate the total by multiplying 330 kg · 35 = 192.5 kg 60
35 elbow pipes weigh
ME
330 kg 60
Three steps for calculating inverse proportional ratios Example:
PE
200 h 150 s r u100 o h
Known data 2nd step:
50 0 0
It takes 3 workers 170 hours to process one order. How many hours do 12 workers need to process the same order? It takes 3 workers 170 hours
Calculate the unit time by multiplying
It takes 1 worker 3 · 170 hrs
2
4 6 8 10 12 14 workers
3rd step:
Calculate the total by dividing
It takes12 workers
3 · 170 hrs = 42.5 hrs 12
Using the three steps for calculating direct and inverse proportions
A
Example: 660 workpieces are manufactured by 5 machines in 24 days.
1st application of 3 steps: 5 machines produce 660 workpieces in 24 days 1 machine produces 660 workpieces in 24 · 5 days 9 machines produce 660 workpieces in
How much time does it take for 9 machines to produce 312 workpieces of the same type?
S
24 · 5 days 9
2nd application of 3 steps: 24 · 5 9 machines produce 660 workpieces in days 9 9 machines produce 1 workpiece in
24 · 5 days 9 · 660
9 machines produce 312 workpieces in
24 · 5 · 312 = 6.3 days 9 · 660
19
Mathematics: 1.4 Symbols, Units
Formula symbols, Mathematical symbols Formula symbols Formula symbol
Meaning
cf. DIN 1304-1 (1994-03) Formula symbol
Meaning
Formula symbol
M
Meaning
Length, Area, Volume, Angle
Πw h s
Length Width Height Linear distance
r, R d, D A, S V
Radius Diameter Area, Cross-sectional area Volume
a, b, g ² l
Planar angle Solid angle Wave length
P
Mechanics m m* m+
r
J p p abs p amb p g
Mass Linear mass density Area mass density Density Moment of inertia Pressure Absolute pressure Ambient pressure Gage pressure
F F W, W M T M b
s t e
E
Force Gravitational force, Weight Torque Torsional moment Bending moment Normal stress Shear stress Normal strain Modulus of elasticity
G f W
μ,
I
W, E W p, E p W k, E k P
n
Shear modulus Coefficient of friction Section modulus Second moment of an area Work, Energy Potential energy Kinetic energy Power Efficiency
TD
Time t T n
Time, Duration Cycle duration Revolution frequency, Speed
f, v v, u
w
Frequency Velocity Angular velocity
a g
a·
Q, V, q v
Acceleration Gravitational acceleration Angular acceleration Volumetric flow rate
Electricity Q E C I
Electric charge, Quantity of electricity Electromotive force Capacitance Electric current
L R
r g, k
Inductance Resistance Specific resistance Electrical conductivity
X Z
j
N
Reactance Impedance Phase difference Number of turns
MS
Heat T, Q
DT, Dt, Dh t, h a —, a
Thermodynamic temperature Temperature difference Celsius temperature Coefficient of linear expansion
Q
l a
k
Heat, Quantity of heat Thermal conductivity Heat transition coefficient Heat transmission coefficient
· G, Q a c H net
Heat flow Thermal diffusivity Specific heat Net calorific value
ME
Light, Electromagnetic radiation E
Illuminance
f n
Focal length Refractive index
LP
Acoustic pressure level Sound intensity
I
Q , W
Luminous intensity Radiant energy
Acoustics p c
Acoustic pressure Acoustic velocity
I
N LN
Mathematical symbols Math. symbol
½ ‡ …
6 =
Ï def
== <
‰ >
›
+ – · –, /, :, ÷ V
Spoken approx. equals, around, about equivalent to and so on, etc. infinity equal to not equal to is equal to by definition less than less than or equal to greater than greater than or equal to plus minus times, multiplied by over, divided by, per, to sigma (summation)
Loudness Loudness level
PE
cf. DIN 1302 (1999-12) Math. symbol
,
an
0 3 0 3 æxæ o ø º º n
º @ ™ 9 Dx º
%
‰
Spoken
Math. symbol
Spoken
proportional a to the n-th power, the n-th power of a square root of n-th root of
log lg ln e
logarithm (general) common logarithm natural logarithm Euler number (e = 2.718281…)
absolute value of x perpendicular to is parallel to parallel in the same direction
sin cos tan cot
sine cosine tangent cotangent
parallel in the opposite direction angle triangle congruent to
(), [], {}
delta x (difference between two values) percent, of a hundred per mil, of a thousand
p AB
£ AB
a*, a+ a1, a2
A
parentheses, brackets open and closed pi (circle constant = 3.14159…) line segment AB arc AB a prime, a double prime a sub 1, a sub 2
S
20
Mathematics: 1.4 Symbols, Units
SI quantities and units of measurement M
P
SI1) Base quantities and base units
cf. DIN 1301-1 (2002-10), -2 (1978-02), -3 (1979-10)
Base quantity
Length
Mass
Time
Electric current
Thermodynamic temperature
Amount of substance
Luminous intensity
Base units
meter
kilogram
second
ampere
kelvin
mole
candela
m
kg
s
A
K
mol
cd
Unit symbol 1)
The units for measurement are defined in the International System of Units SI ( Système International d’Unités). It is based on the seven basic units (SI units), from which other units are derived.
Base quantities, derived quantities and their units Quantity
TD
Symbol
Relationship
Remarks Examples of application
= 10 dm = 100 cm = 1000 mm 1 mm = 1000 µm 1 km = 1000 m
1 inch = 25.4 mm In aviation and nautical applications the following applies: 1 international nautical mile = 1852 m
Length, Area, Volume, Angle Length
Area
Œ
A, S
MS Volume
ME
Unit Name Symbol
Plane angle (angle)
Solid angle
V
meter
m
1m
square meter
m2
1 m2
are hectare
a ha
cubic meter
m3
liter
—, L
a, b, g … radian
rad
Symbol S only for cross-sectional = 10 000 cm2 areas = 1 000 000 mm2 2 1a = 100 m 1 ha = 100 a = 10 000 m2 Are and hectare only for land 100 ha = 1 km2 1 m3
= 1000 dm3 = 1 000 000 cm3 1 — = 1 L = 1 dm3 = 10 d — = 0.001 m3 1 m — = 1 cm3 1 rad = 1 m/m = 57.2957…° = 180°/ p p 1° = rad = 60* 180
degrees
°
minutes
*
1*
= 1°/60 = 60+
seconds
+
1+
* = 1 /60 = 1°/3600
≈
steradian
sr
1 sr
= 1 m2 /m2
m
kilogram gram
kg g
1 kg 1g
megagram metric ton
Mg t
PE
Mostly for fluids and gases
1 rad is the angle formed by the intersection of a circle around the center of 1 m radius with an arc of 1 m length. In technical calculations instead of a = 33° 17* 27.6+, better use is a = 33.291°. An object whose extension measures 1 rad in one direction and perpendicularly to this also 1 rad, covers a solid angle of 1 sr.
Mechanics Mass
A
1 metric t = 1000 kg = 1 Mg 0.2 g = 1 ct
Mass in the sense of a scale result or a weight is a quantity of the type of mass (unit kg).
Mass for precious stones in carat (ct).
Linear mass density
m*
kilogram per meter
kg/m
1 kg/m = 1 g/mm
For calculating the mass of bars, profiles, pipes.
Area mass density
m+
kilogram per square meter
kg/m2
1 kg/m2 = 0.1 g/cm2
To calculate the mass of sheet metal.
kilogram per cubic meter
kg/m3
1000 kg/m3 = 1 metric t/m 3 = 1 kg/dm3 = 1 g/cm3 = 1 g/ml = 1 mg/mm3
The density is a quantity independent of location.
Density
S
= 1000 g = 1000 mg
r
